Ch 28: Fundamentals of Circuits

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 28: Fundamentals of Circuits

Problem 43

Chapter 28, Problem 43

You have a 2.0 Ω resistor, a 3.0 Ω resistor, a 6.0 Ω resistor, and a 6.0 V battery. Draw a diagram of a circuit in which all three resistors are used and the battery delivers 9.0 W of power.

Verified step by step guidance

Step 1: Recall the formula for power delivered by a battery: $ P = \frac{V^2}{R_{eq}} $, where $ P $ is the power, $ V $ is the voltage of the battery, and $ R_{eq} $ is the equivalent resistance of the circuit. Rearrange this formula to find $ R_{eq} $: $ R_{eq} = \frac{V^2}{P} $. Substitute $ V = 6.0 \ \text{V} $ and $ P = 9.0 \ \text{W} $ to calculate $ R_{eq} $.

Step 2: Once $ R_{eq} $ is determined, consider how the three resistors (2.0 Ω, 3.0 Ω, and 6.0 Ω) can be arranged to achieve this equivalent resistance. Recall the formulas for resistors in series $ R_{eq} = R_1 + R_2 + R_3 $ and in parallel $ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} $.

Step 3: Experiment with different combinations of series and parallel arrangements to match the calculated $ R_{eq} $. For example, you might try combining two resistors in parallel and then adding the third in series, or vice versa. Use the formulas for series and parallel resistances to verify the equivalent resistance for each configuration.

Step 4: Once the correct arrangement is identified, draw the circuit diagram. Ensure that the battery (6.0 V) is connected to the resistors in the configuration that achieves the desired $ R_{eq} $. Label the resistors and their values (2.0 Ω, 3.0 Ω, and 6.0 Ω) clearly in the diagram.

Step 5: Double-check your work by recalculating the total power delivered by the battery using $ P = \frac{V^2}{R_{eq}} $ with the equivalent resistance from your chosen configuration. Confirm that the power matches the given value of 9.0 W.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship is expressed by the formula V = IR, which is fundamental for analyzing electrical circuits and determining how voltage, current, and resistance interact.

Power in Electrical Circuits

The power (P) delivered by an electrical circuit is defined as the rate at which energy is consumed or converted. It can be calculated using the formula P = IV, where I is the current and V is the voltage. Additionally, using Ohm's Law, power can also be expressed as P = I²R or P = V²/R, which helps in understanding how different components in a circuit affect overall power consumption.

Series and Parallel Circuits

In electrical circuits, resistors can be arranged in series or parallel configurations. In a series circuit, the total resistance is the sum of individual resistances, while in a parallel circuit, the total resistance is found using the reciprocal formula. Understanding these configurations is crucial for calculating the total resistance and current in the circuit, which directly impacts the power delivered by the battery.

Recommended video:

Guided course

09:51

Combining Capacitors in Series & Parallel

Related Practice

Textbook Question

A small toaster that operates at 120 V has a heating element made from a 4.4-m-long, 0.70-mm-diameter nichrome wire. The resistivity, density, and specific heat of nichrome are, respectively, 1.5 x 10⁻⁶ Ωm, 8400 kg/m³, and 450 J/kg K. If half the heat energy is lost to the air, how long does it take the heating element to warm from 20℃ to 450℃, about the temperature at which it first begins to glow red?

views

Textbook Question

It seems hard to justify spending \$5.00 for an LED lightbulb when an ordinary incandescent bulb costs 50¢. To see if this makes sense, compare a 60 W incandescent bulb that lasts 1000 hours to a 10 W LED bulb that has a lifetime of 15,000 hours. Both bulbs produce the same amount of visible light. If electricity costs \$0.15/kWh, what is the total cost—purchase price plus energy—to get 15,000 hours of light from each type of bulb? This is called the life-cycle cost.

views

Textbook Question

A capacitor is discharged through a 100 Ω resistor. The discharge current decreases to 25% of its initial value in 2.5 ms. What is the value of the capacitor?

views

Textbook Question

A 2.0-m-long, 1.0-mm-diameter wire has a variable resistivity given by where x is measured from one end of the wire. What is the current if this wire is connected to the terminals of a 9.0 V battery?

views

Textbook Question

An electric eel develops a 450 V potential difference between its head and tail. The eel can stun a fish or other prey by using this potential difference to drive a 0.80 A current pulse for 1.0 ms. What are (a) the energy delivered by this pulse and (b) the total charge that flows?

views

Textbook Question

To which two points in the circuit of FIGURE P28.45 should a 12 V battery be connected to dissipate the most power?

views